2D Spectral Element Scheme for Viscous Burgers' Equation

2D Spectral Element Scheme for Viscous Burgers’ Equation 

P. Aaron Lott 

University of Maryland 

Department of Applied Math & 

Scientific Computation 

AMSC 664 

May 14, 2004 

Advisor: Dr. Anil Deane

AMSC 664: 2D Spectral Element Scheme for Viscous Burgers’ Equation 2 

Motivation/Scientific Context 

● To model the dynamics of the Earth’s Mantle we treat it as a highly 

viscous, incompressible Boussinesq fluid. 

● It is important to study the affects of flows over long time periods 

to better constrain the parameter space of the model. 

● There is seismic and geochemical evidence of chemical/structural 

phase transition at the depth of 410 and 670 km. There are viscosity 

changes of several orders of magnitude. To handle these sharp 

interfaces one needs a refinement method to efficiently study the 

flow in this region. 

P. Aaron Lott palott@ipst.umd.edu


● The Governing Equations are: 

∇ · u = 0 Incompressibility 

Mantle Convection 

1 

ρ∇p = ν∇2u − gαΔT (Momentum Equation with∂u ∂t 

= 0) 

∂T 

∂t + u · ∇T = κ∇2T + J ( Thermal energy equation) 

ρCp 

● u - velocity, T - temp, p is pressure, ν -viscosity, κ - thermal 

diffusivity, α - thermal expansion coefficient, ρ - density, and Cp - 

heat capacity at constant pressure, J - rate of internal pressure per 

unit volume, g - gravity. 

● Note, these are the Incompressible Steady Stokes Equations with 

a source term fed by the unsteady advection diffusion equation at 

each time step. 



Advection Diffusion Equation 

● We are solving the Advection Diffusion equations in 1D and 2D, 

with advection velocityc and viscosity ν. 

● 1D 

● 2D 

∂u 

∂t 

∂u 

∂t 

+ (c∂u 

∂x ) = ν∂2 u 

∂x2 (1) 

= νΔu −c · ∇u in Ω t ≥ 0 (2) 

● Notec = u yields the viscous Burgers’ Equations. 



Spectral Element Discretization 

● We use a Spectral Element Method Discretization to solve 1 and 2, 

expanding the solution as a linear combination of basis functions 

φi(x). 

u k n = 

n 

∑ u 

i=0 

k i (t)φi(x) (3) 



● 

Spatial Discretization- Basis Functions 

Figure 1: A. GLL Spatial Discretization with 2 elements and 5th degree Polynomials B. GLL Polynomials of degree 1 

through 10 



Spatial Discretization 

● The discretized equation can be written as a non-linear differential 

equation 

M ˙u(t) = −C(u)u(t) − νKu(t) (4) 

● M- Mass matrix, K- Stiffness matrix and C(u) is the nonlinear 

discrete operator. Each of which are block diagonal matrices. 

● So to solve for u(x,t) we will integrate these equations in time. 



● 

Spatial Discretization 

Figure 2: Global System Matrices. From the left 1 Mass,1, Convection, 1 Stiffness(Diffusion) 



Time Discretization 

● In order to obtain a stable solution in time, one considers the 

eigenvalues of the operators acting on u, and makes certain that 

the time marching scheme is stable in this region. 

● In our system, C and K act on u 

● 

Figure 3: Eigenvalues of the Diffusion and Convection operators 




● For spectral methods the eigenvalues, λ, of the diffusion matrix 

are real and negative, and the maximum eigenvalue is O(N 4 ) where 

N is the maximum polynomial degree. For Spectral Elements, 

empirical tests show λ ≈ O(neN 3 ) where ne is the number of elements. 

● The eigenvalues, λ, of the convection operator have an imaginary 

part and and a negative real part, and the largest eigenvalue is 

O(N 2 ). 

Thus, we want a time discretization which is stable on the negative 

real axis and the imaginary axis. 



● 


Figure 4: Stability region arrows denote stability outside the corresponding curve for the Backward Difference time 

marching scheme. 




● In order to achieve 3rd order accuracy, and a stable solution in 

time we use the BDF3 scheme, and extrapolate the Convection 

term at each iteration. 

( 11 

M 

M + νK)vn+1 i = 

6Δt Δt (3vni − 3 

2 vn−1 

1 

i + 

3 vn−2 i ) −Cv n+1 

i 

where we use 3rd order extrapolation to obtain 

Cv n+1 

i 

= 3Cvn i − 3Cv n−1 

i 

(5) 

+Cvn−2 

i + O(Δt3 ) (6) 

● For SEM a harsh condition is placed on Δt in order to satisfy the 

CFL criteria. For basis functions of degree N − 1, 

Δt ≤ 6.5 π 

ν 

2 

N4 (7) 



Computation Localization 

● We originally formed the global system matrices, and then iterated 

over time. However, as we moved into 2D these system matrices 

have size (P + 1) 2 NxNy, which, even for coarse meshes are quite 

large. 

● Static Condensation 

Earlier we gave an illustration of the coupling between elements. 

By re-ordering the mapping between local and global indices we 

could effectively decouple the interiors and only solve the coupled 

system on elemental boundaries. This could all be done with local 

element matrices of size (P + 1) 2 . 



● 


Figure 5: Coupled and Uncoupled Diffusion operator. 1 Big coupled 1 Boundary 1Interior 




● Advantages: 

Reduces communication once the boundary information has been 

decoupled. No global matrices need to be stored, all calculations 

are done on element matrices 

● Disadvantages: Need to compute Schur compliment. 




● Elementwise operations 

Instead of using static condensation, one can perform operations 

on local elements and then cleverly add the proper amount to the 

global solution U without the cost of static condensation. 

● A weighting matrix is formed for each element to determine the 

contribution of the local solution to the global solution. Finally we 

take care of the element boundary dependencies after each local 

calculation. 




● Advantages: For large P this should be very efficient since most 

entries are interior nodes, and each processor can use its cache 

more efficiently. No global matrices need to be stored, all calculations 

are done on element matrices 

● Disadvantages: For small polynomial degree, more communication 

compared to calculations 




● In either case, these problems scale well to higher dimensions because 

the global Matrix operators can be written as tensor products. 

● We construct the local operators for all possible polynomial degree 

(run time parameter), and store them in an easily accessible 

efficient data structure. These include, M,A,C, Derivative, Iterpolants 

from Pn → Pn−1 and vice versa. These structures are accessible 

through a global data module. 



P-type Refinement 

● With the local matrices stored for all values of P, and the ability 

to perform local operations and build the global solution, it is now 

trivial to compute the derivative of the local solution and perform 

error analysis with it. 

● For example, if the slope of our solution at a local element is 

greater than some user defined value, then we increase the polynomial 

degree of that element by one. 

● We perform this on each element and then construct a new local 

to global mapping with respect to the new elements. 



● 

P-type Refinement 

Figure 6: 3 time steps in solution to burgers’ equation starting with N=32 P=4 refining with abs(du/dx) is greater 

than 8 



● 1D Burgers’s Equation 

Validation 

Figure 7: Comparison of Results between Our code (left) and Published Code (right) 



● 

Validation 

Figure 8: Comparison of results between our code and actual maximum amplitude of slope. We have a value of 

152.2265 at t=.5100 analytical value is 152.0051 at t=.5105. The compares very well with other published numerical 

methods. 



● 2D Burgers’s Equation 

Start with initial conditions 

2D Results 

u(x,y,0) = .01 4(x2 +y 2 ) 

on[−1,1] 2 

We use periodic boundary conditions, P = 8, Nx = 4, Ny = 4 ν = 

.01 

P. Aaron Lott palott@ipst.umd.edu 

(8)


● Add 2D adaptivity 

● Implement Stokes Equation 

● Parallelization 

Future Directions 

● Preconditioned Conjugate Gradient to solve local systems 



Conclusions/Summary 

● Implemented and verified 1D and 2D viscous Burgers’ Equation 

● Implemented P adaptivity in 1D, and framework for adaptivity in 

2D 

● Structured code to begin MPI Parallelization 



References 

● [1] High-Order Methods for Incompressible Fluid Flows. M.O. 

Deville, P.F. Fischer, E.H. Mund. Cambridge University Press 

2002. 

[2] An overlapping Schwarz method for spectral element solution 

of the incompressible Navier-Stokes equations. P.F. Fischer. Journal 

of Computational Physics 1997. 

[3] Spectral/hp Element Methods for CFD. G.E. Karniadakis, S.J. 

Sherwin. Oxford University Press 1999. 

[4] An Adaptive Finite Element Scheme for Transient Problems 

in CFD. Rainald Löhner. Computational Methods in Applied Mechanics 

and Engineering. 61 (1987) 323-338 

[5] Spectral element methods: theory and applications. F.N. van 

de Vosee, P.D. Minev. 

http://www.mate.tue.nl/people/vosse/docs/vosse96b.pdf


[6] Project Website. http://www.lcv.umd.edu/ palott/research/graduate/66 



● 2 

Figures 



● 2 

Figures 



● 2 

Figures 



● 5 

Figures 



● 5 

Figures 



● 5 

Figures 

The End — Thank you

2D Spectral Element Scheme for Viscous Burgers' Equation

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?